{"paper":{"title":"Boundedness of the extremal solutions in dimension 4","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Salvador Villegas","submitted_at":"2012-06-27T11:30:50Z","abstract_excerpt":"In this paper we establish the boundedness of the extremal solution u^* in dimension N=4 of the semilinear elliptic equation $-\\Delta u=\\lambda f(u)$, in a general smooth bounded domain Omega of R^N, with Dirichlet data $u|_{\\partial \\Omega}=0$, where f is a C^1 positive, nondecreasing and convex function in [0,\\infty) such that $f(s)/s\\rightarrow\\infty$ as $s\\rightarrow\\infty$.\n  In addition, we prove that, for N>=5, the extremal solution $u^*\\in W^{2,\\frac{N}{N-2}}$. This gives $u^\\ast\\in L^\\frac{N}{N-4}$, if N>=5 and $u^*\\in H_0^1$, if N=6."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.6233","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}